A large part of jazz harmony is based on the major scale. As discussed earlier, every major scale has a relative minor that is formed by playing the same sequence of notes but starting on the sixth step of the scale. In fact, a scale can be formed using the sequence of notes from a major scale starting on any step of the scale. These scales are called modes of the scale. The major scale itself is called the ionian mode. The sixth mode, the relative minor, is called the aeolian mode. The names of these modes, as well as the others discussed below, come from ancient Greece, although the names are rumored to have been mixed up in translation long ago. While the Greek modes are mainly only of historical interest in classical theory, they are fundamental to jazz.
The major scale, or ionian mode, should be quite familiar by now. It is associated with major seventh chords. In the key of C, for example, the C major seventh chord, notated Cmaj7 (or C with a little triangle next to it, or sometimes CM7), is "C E G B", and these notes outline the C major scale. If a measure in a piece of music is harmonized with a Cmaj7 chord, then the C major scale is one appropriate scale to use when improvising. The only note in this scale that sounds bad when played against a Cmaj7 chord is the fourth note, F. You may wish to convince yourself of this by going to a piano and playing Cmaj7 in your left hand while playing various notes from the C major scale in your right. The fourth of the major is often called an avoid note over a major seventh chord. This does not mean you are not allowed to ever play F over a Cmaj7, of course, but you should be conscious of the dissonant effect it produces.
The chord obtained by adding another third on top ("C E G B D") would be called a Cmaj9, and it implies the same scale. Adding another third on top would yield "C, E, G, B, D, F", and this chord would be called a Cmaj11. Because of the dissonant nature of the F in this context, however, neither this chord, nor the Cmaj13 chord obtained by adding an additional third (A), are used very much.
The dorian mode is built on the second step of the major scale, using the same notes. For example, the D dorian scale is built from the notes of the C major scale, starting on D, and consists of "D, E, F, G, A, B, C". The dorian mode is a lot like minor scale, but the sixth step is raised a half step. That is, the D minor scale would have a Bb while the dorian has a B. Because it is so similar to the minor scale, it is natural to play this scale over a minor seventh chord. In fact, it is used more often than the minor scale itself. If you go to a piano and play a Dm7 chord ("D F A C") in your left hand, and play notes from the D dorian and D minor scales in your right, you will probably find that the dorian mode sounds better, because the B is less dissonant against the Dm7 than the Bb is. If you use the dorian mode over a minor seventh chord, there are no notes to avoid.
Like the major seventh chord, you can add more thirds to the minor seventh chord to obtain Dm9, Dm11, and Dm13. These chords still imply the same dorian mode. If you use the natural minor scale, the thirteen chord contains the note Bb, which is somewhat dissonant in this context. This chord is seldom used, but when it is called for, it is often notated Dm7b6, and is one of the few exceptions to the rule that most chords are written in terms of odd numbered extensions above the seventh. This rule comes from the fact that chords are traditionally built by stacking thirds. The notation Dm6 is sometimes as a synonym for Dm13 when the B natural is explicitly meant.
The third mode of the major scale is called the phrygian mode. In the key of C, a phrygian scale is built on E, and consists of "E, F, G, A, B, C, D". This scale, like the dorian mode, is also similar to the minor scale, except that the second step in the phrygian mode is lowered by a half step. That is, an E minor scale would have an F# while the phrygian has an F. If you try playing the phrygian scale over a minor seventh chord, you will probably find it more dissonant than the minor scale, because of the lowered second. The phrygian mode is used occasionally over a minor seventh chord, although often the chord is written as m7b9 as a hint to the improviser that the phrygian scale is to be used. There are certain other situations in which the phrygian scale sounds good. One is over a dominant seventh chord with a suspended fourth (see mixolydian mode, below) and a lowered ninth, notated susb9. Another is over a particular chord that I will simply call a phrygian chord. A phrygian chord in E would be "E F A B D". When the phrygian mode is played over this type of chord, the result is a somewhat Spanish sound, particularly if you add a G# to the scale, yielding what is sometimes called the Spanish phrygian scale. Several Chick Corea tunes, including "La Fiesta", and much of the music from Miles Davis' Sketches Of Spain feature this sound extensively.
The fourth mode of the major scale is the lydian mode. In the key of C, a lydian scale is built on F, and consists of "F, G, A, B, C, D, E". This scale is like the major scale except that it contains a raised fourth step. That is, an F major scale would contain a Bb while the lydian contains a B. Since the fourth step of the major scale is an avoid note over a major seventh chord, this scale gives the improviser an alternative. While the raised fourth might sound a little unusual at first, you should find that it is in general preferable to the natural fourth of the major scale. When the symbol Cmaj7 appears, you have a choice between the major and lydian scales. Often, if the lydian mode is specifically intended, the symbol Cmaj7#11 will appear instead. Recall that Cmaj11 contains an F as the eleventh; Cmaj7#11 denotes that this note should be raised by a half step.
The fifth mode of the major scale is the mixolydian mode. In the key of C, a mixolydian scale is built on G, and consists of "G, A, B, C, D, E, F". This scale is like the major scale except that the seventh step is lowered a half step. That is, a G major scale would contain an F# while the mixolydian contains an F. Since the seventh chord built on the fifth degree of the major scale is a dominant seventh, it is natural to play lines based on the mixolydian mode over a dominant seventh chord. For instance, the G mixolydian scale might be used over a G7 chord.
As with the major scale over a major seventh chord, the fourth step of the scale (C in the case of G mixolydian) is somewhat of an avoid note over a dominant seventh chord. However, there is a chord called a suspended chord, notated Gsus, Gsus4, G7sus, G7sus4, F/G, Dm7/G, or G11 over which there are no avoid notes in the G mixolydian mode. The notation F/G indicates an F major triad over the single note G in the bass. The term "suspension" comes from classical harmony and refers to the temporary delaying of the third in a dominant chord by first playing the fourth before resolving it to the third. In jazz, however, the fourth often is never resolved. The suspended chord consists of the root, fourth, fifth, and usually the seventh as well. Herbie Hancock's tune "Maiden Voyage" consists solely of unresolved suspended chords.
The aeolian mode, or minor scale, has already been discussed. It can be played over a minor seventh chord, although the dorian or phrygian modes are used more often. It is most often played over a m7b6 chord.
The seventh and final mode of the major scale is the locrian mode. In the key of C, a locrian scale is built on B, and consists of "B, C, D, E, F, G, A". The seventh chord built on this scale ("B D F A") is a half diminished seventh chord, Bm7b5. This symbol comes from the fact that this chord is similar to a Bm7, except that the fifth is lowered by a half step. The classical symbol for this chord is a circle with a "/" through it. The locrian scale can be used over a half diminished (also called a minor seven flat five) chord, but the second step is somewhat dissonant and is sometimes considered an avoid note.